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The volume of the Lp-centroid body of a convex body K ⊂ ℝd is a convex function of a time-like parameter when each chord of K parallel to a fixed direction moves with constant speed. This fact is used to study extrema of some affine invariant functionals involving the volume of the Lp-centroid body and related to classical open problems like the slicing problem. Some variants of the Lp-Busemann-Petty centroid inequality are established. The reverse form of these inequalities is proved in the two-dimensional case.

The notion of generalized X-ray for star sets in a Riemannian manifold is introduced to prove uniqueness theorems for convex bodies contained in a simply convex neighbourhood of a two-manifold. These results extend to the whole space and to arbitrary dimension when spaces of constant curvature are considered. As a consequence, a characterization of centrally symmetric convex bodies is obtained.

Absolute curvature measures for locally finite unions of sets with positive reach are introduced, extending the definition of Zähle [13] by taking into account the absolute value of the index function. It is shown that this definition differs from that of Matheron [5] and Schneider [12]. An intersection formula of Crofton type for absolute curvature measures is proved. The role of absolute curvature measures in geometric statistics is illustrated by an example.

Let K be a compact convex body in ℝn not contained in a hyperplane, and denote the norm whose unit ball is ½(K − K) by ║·║k. Given a translative packing of K, we are interested in how long a segment (with respect to ║·║K) can lie in the complement of the interiors of the translates. The main result of this note is to show the existence of a translative packing such that the length of the longest segments avoiding it is only exponential in the dimension n (see below). We start here with a lower bound, showing that this bound is close to optimal for balls.

All those multiplications on the two-dimensional Euclidean group are determined such that the resulting non-associative topological nearring has (1, 0) for a left identity and has the additional property that every element of the near-ring is a right divisor of zero. This result, together with several previous results, is then used to show that any one of several common algebraic properties is sufficient to characterize one particular two-dimensional Euclidean ring within the class of all two dimensional Euclidean near-rings. Specifically, it is proved that, if N is a topological near-ring with a left identity whose additive group is the two-dimensional Euclidean group, then the following assertions are equivalent: (1) the left identity is not a right identity, (2) N contains a non-zero left annihilator, (3) every element of N is a right divisor of zero, (4) Nw≠N for all w∈N, (5) N is isomorphic to the topological ring whose additive group is the two dimensional Euclidean group and whose multiplication is given by (v1, V2)(w1W2) = (v1w1, v1w2).

Let R be a Noetherian local ring with maximal ideal m and lull ring of fractions Q. In this paper we consider a numerical function EHI: ℤ → ℤ, where I is an m-primary ideal of R, that coincides with the Hilbert function HI for positive values and that takes account of the fractional powers of I for negative values. We focus our attention on the one-dimensional case. Among other results we characterize one-dimensional Gorenstein local rings by means of the symmetry of EHR in Theorem 2.1, we show that the extended Hilbert function is not determined by the Hilbert function in Example 2.2. and we generalize to m-primary ideals the upper bound for e1(m) given by Matlis for the maximal ideal.

This paper treats rational homology groups of one-point compactifications of spaces of complex monic polynomials with multiple roots. These spaces are indexed by number partitions. A standard reformulation in terms of quotients of orbit arrangements reduces the problem to studying certain triangulated spaces Xλ,μ.

In this paper, further insight is obtained into the earlier approach of studying residually transcendental extensions of a valuation v of a field K to a simple transcendental extension K(x) of K by means of minimal pairs, thereby introducing new invariants corresponding to any element of an algebraic closure of K. It is also shown that these invariants are of independent interest as well. A characterization of those elements a belonging to is given such that there exists a minimal pair (a, δ) for some δ in the divisible closure of the value group of v.

Let v be a Henselian valuation of any rank of a field K and its unique prolongation to a fixed algebraic closure of K having value group . For any subfield L of , let R(L) denote the residue field of the valuation obtained by restricting to L. Using the canonical homomorphism from the valuation ring of v onto its residue field R(K), one can lift any monic irreducible polynomial with coefficients in R(K) to yield a monic irreducible polynomial with coefficients in K. In an attempt to generalize this concept, Popescu and Zaharescu introduced the notion of lifting with respect to a (K, v)-minimal pair (α, δ) belonging to × . As in the case of usual lifting, a given monic irreducible polynomial Q(y) belonging to R(K(α))[y] gives rise to several monic irreducible polynomials over K which are obtained by lifting with respect to a fixed (K, v)-minimal pair (α, δ). If F, F1 are two such lifted polynomials with coefficients in K having roots θ, θ1, respectively, then it is proved in the present paper that in case (K, v) is a tame field, it is shown that K(θ) and K(θ1) are indeed K-isomorphic.

This paper concerns the spectrum of the r-dimensional Sturm–Liouville equation y″ + (λI − Q(x))y = 0 with the Dirichlet boundary conditions, where Q is an r × r symmetric matrix. It is proved that, under certain conditions on Q, this problem can only have a finite number of eigenvalues with multiplity r. Further discussion is given for the multiplicities of eigenvalues when Q is an r × r Jacobian matrix.

Existence criteria are presented for non-linear boundary value problems on the half line. In particular, the theory includes a problem in the theory of colloids and a problem arising in the unsteady flow of a gas through a semi-infinite porous medium.

The main goal of this paper is to prove that the classical theorem of local inversion for functions extends in finite dimension to everywhere differentiable functions. As usual, a theorem of implicit functions can be deduced from this “Local Inversion Theorem”. The deepest part of the local inversion theorem consists of showing that a differentiable function with non-vanishing Jacobian determinant is locally one-to-one. In turn, this fact allows one to extend the Darboux property of derivative functions on ℝ (the range of the derivative is an interval) to the Jacobian function Df of a differentiable function, under the condition that this Jacobian function does not vanish. It is also proved that these results are no longer true in infinite dimension. These results should be known in whole or part, but references to a complete proof could not be found.

It is shown that Lipschitz quotient mappings between finite dimensional spaces behave nicely (e.g., are bijective in the case of equal dimensions) if the Lipschitz and co-Lipschitz constants are close to each other. For Lipschitz quotient mappings of the plane, a bound for the cardinality of the pre-image of a point in terms of the ratio of the constants is obtained.

A diophantine system is studied from which is deduced an analogue of van der Corput's A5-process in order to bound analytic exponential sums of the form . The saving has now to be taken to the exponent 1/20 instead of 1/32. Our main application is a “ninth derivative test” for exponential sums which is essential for giving new exponent pairs in [3].

For an indefinite quadratic form f(x1,…,xn) of discriminant d. let P(f) denote the greatest lower bound of the positive values assumed by f for integers x1, …, xn. This paper investigates the values of P3/|d| for nonzero ternary forms of signature −1, and finds the only remaining class of forms with .

A positive definite integral quadratic form over rational integers is said to be universal, if it represents all positive integers. The universal quaternary quadratic form is determined with the maximal discriminant, which is 1073/4.

Let L be a Galois extension of the number field K. Set n = nK = deg K/ℚ, nL = deg L/ℚ and nL/K = deg L/K. Let I = IL/K denote the group of fractional ideals of K whose prime decomposition contains no prime ideals that ramify in L, and let P = {(α)ΣI: αΣK*, α>0}. Following Hecke [9}, let (λ1, λ2, …, λn − 1) be a basis for the torsion-free characters on P that satisfy λi(α) = 1 (1≤i≤n − 1) for all units α>0 in , the ring of integers of K. Fixing an extension of each λi to a character on I, then λi,(α) (1 ≤i≤n − 1) are defined for all ideals α of K that do not ramify in L. So, for such ideals, we can define . Then the small region of K referred to above is

for 0<l<½ and with the notation that, for any α ∈ ℝ, we set β. where β is the unique real satisyfing .

Fermat gave the first example of a set of four positive integers {a1, a2, a3, a4} with the property that aiaj+1 is a square for 1≤i<j≤4. His example was {1, 3, 8, 120}. Baker and Davenport [1] proved that the example could not be extended to a set of 5 positive integers such that the product of any two of them plus one is a square. Kangasabapathy and Ponnudurai [6], Sansone [9] and Grinstead [4] gave alternative proofs. The construction of such sets originated with Diophantus who studied the problem when the ai are rational numbers. It is conjectured that there do not exist five positive integers whose pairwise products are all one less than the square of an integer. Recently Dujella [3] proved that there do not exist nine such integers. In this note we address the following related problem. Let V denote the set of pure powers, that is, the set of positive integers of the form xk with x and k positive integers and k>1. How large can a set of positive integers A be if aa′ + 1 is in V whenever a and a′ are distinct integers from A? We expect that there is an absolute bound for |A|, the cardinality of A. While we have not been able to establish this result, we have been able to prove that such sets cannot be very dense.